Nonlinear optical frequency conversion using metamaterial arrays

ABSTRACT

A method of nonlinear wavelength generation uses a nonlinear optical medium. An input flux of pump energy is applied to one or more dielectric optical resonators. Each resonator has an optical cavity comprising the nonlinear optical medium. Each resonator has at least one Mie resonance that is excited by the input flux of pump energy. The pump energy causes the generation of converted light containing at least one converted component having a frequency attainable only through a non-linear process.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.62/351,111, filed Jun. 16, 2016, the entirety of which is herebyincorporated herein by reference.

This application is a continuation-in-part of U.S. application Ser. No.15/184,929 filed by Sheng Liu et al. on Jun. 16, 2016 under the title“Optoelectronic Apparatus Enabled by Dielectric Metamaterials”, theentirety of which is hereby incorporated herein by reference.

The abovesaid U.S. application Ser. No. 15/184,929 claims the benefit ofthe earlier filing date of U.S. Provisional Patent Application No.62/182,381, filed Jun. 19, 2015, titled “Optoelectronic ApparatusEnabled by Dielectric Metamaterials”.

STATEMENT OF GOVERNMENT RIGHTS

This invention was developed under Contract DE-AC04-94AL85000 betweenSandia Corporation and the U.S. Department of Energy and under ContractDE-NA0003525 awarded to National Technology & Engineering Solutions ofSandia, LLC by the United States Department of Energy/National NuclearSecurity Administration. The U.S. Government has certain rights in thisinvention.

FIELD OF THE INVENTION

The invention relates to optical devices, and more particularly tooptical devices that utilize metamaterials.

ART BACKGROUND

Nonlinear optics is the study of how intense laser light interacts withoptical materials. Nonlinear optical processes typically generatecoherent photons with new frequencies and wavelengths. Hence one benefitis access to spectra that are not available using conventional lasers.

Bulk nonlinear crystals (generally, crystals that are hundreds ofmicrometers to several millimeters in their spatial dimensions) havebeen widely used for nonlinear optical processes such as second harmonicgeneration, third harmonic generation, sum frequency generation,difference frequency generation, and the like. Despite the wideavailability of nonlinear crystals, there are demands for more compactstructures to realize high efficiency nonlinear processes. Moreover,traditional nonlinear optical processes using bulk crystals requirephase matching between the fundamental frequencies and new generatedfrequencies.

Generally, the use of uniaxial or biaxial bulk nonlinear crystals, orthe application of quasi-phase matching techniques, is required in orderfor the interacting optical fields to meet the strict phase-matchingconditions.

Recently, however, advances in nanostructured optical materials,plasmonics, and metasurfaces have enabled nonlinear optical processesthat do not depend on phase matching. These approaches create tightconfinement and large resonant enhancement of electromagnetic fields,which generate much higher nonlinear efficiencies than in theconstituent materials.

Moreover, metasurfaces comprising arrays of Mie dielectric resonatorshave attracted recent attention at optical frequencies due to their muchlower loss compared with their metallic counterparts. In particular,silicon has been used extensively as the constituent material forall-dielectric metamaterials that have been used for a variety ofapplications including high efficiency Huygens metasurfaces, beamsteering, ultra-thin waveplates, zero-index directional emission andpolarization insensitive holograms.

In the last few years, it was realized that dielectric nanoresonatorscan also be used to greatly enhance nonlinear optical phenomena, due tothe largely enhanced electromagnetic fields inside the resonators andthe larger mode volume. However, due to the centrosymmetric crystalstructure of silicon, second-order nonlinear optical phenomena were notobserved in silicon-based metasurfaces.

Therefore, there has been a need for dielectric metasurfaces based uponother materials that exhibit an intrinsic second order nonlinearsusceptibility (χ⁽²⁾) for fuller exploitation of this approach forenhanced harmonic generation and other second-order nonlinear phenomena.

SUMMARY OF THE INVENTION

Nanoscale resonators made from III-V semiconductors can fulfill theserequirements.

We have demonstrated resonantly enhanced second-harmonic generation(SHG) using dielectric metasurfaces that are made from gallium arsenide(GaAs) which possesses a large intrinsic second-order nonlinearity ofd₁₄˜200 pm/V. We observed the second-harmonic (SH) response from GaAsnano-resonator arrays over a broad spectral range that encompasses boththeir electric and magnetic dipole resonances.

At both resonances, we observed enhanced SHG that is orders of magnitudestronger than the SHG from unpatterned bulk GaAs. Most interestingly,the conversion efficiency at the magnetic dipole resonance was about 100times higher than the conversion efficiency at the electric dipoleresonance. This was attributed, in part, to the increased absorption ofGaAs at the shorter wavelength of the electric dipole resonance.

We also observed that the polarization of the SHG at the magnetic dipoleresonance was orthogonal to the polarization expected of an SHG processmediated by the bulk nonlinearity of GaAs, suggesting an important roleof surface nonlinearities in this class of dielectric metasurfaces.

Our investigations not only improved our understanding of nonlinearoptical processes in these nanostructured materials, but alsohighlighted the opportunities for nonlinear frequency up- anddown-conversion without phase-matching, as well as entangled photon pairgeneration.

We have developed ultra-compact nanostructures that require no phasematching condition and can perform nonlinear processes with highefficiencies. The nanostructures consist of single- or multi-layerarrays of dielectric resonators made of semiconductors with highnonlinear coefficients. (Below, we also refer to the resonatorstructures as “microresonators” or “nanoresonators”. These terms, aswell as the term “resonators” in the present context, are to beunderstood as equivalent.)

More specifically, we contemplate using III-V semiconductors withnon-centrosymmetry and high nonlinear coefficients. Suitable wafers canbe epitaxially grown using MOCVD, MBE, or the like, and thenanostructures can then be defined by standard lithography techniques.

The nonlinear optical processes are performed around resonances of thesemiconductor nanostructures where the optical field is much enhanced.

Due to this large enhancement inside the semiconductor resonators, theefficiency of the nonlinear optical processes is largely increased.Moreover, the ultra-small dimensions of each resonator eliminate theneed for phase matching.

Accordingly, an embodiment of the invention is a method of nonlinearwavelength generation. An input flux of pump energy is applied to one ormore dielectric optical resonators. For example, a plurality ofnominally identical such resonators may be deployed in an array to whichthe pump energy is applied. “Nominally identical” means identical towithin manufacturing tolerances. Each resonator has an optical cavitycomprising a nonlinear optical medium, and has at least one Mieresonance that is excited by the input flux of pump energy. Applicationof the pump energy causes the generation of converted light containingat least one converted component having a frequency attainable onlythrough a non-linear process. A beam of output light comprising the atleast one converted component is collected from the one or moredielectric resonators.

In some embodiments, the nonlinear optical medium is a bulk material. Inother embodiments, it comprises a semiconductor host and a quantum wellmultilayer embedded in the semiconductor host.

In embodiments, the at least one Mie resonance comprises a magneticdipole resonance or an electric dipole resonance or both types ofresonance. The resonators may also have resonances of higher order thanthe electric and magnetic dipole resonances. Accordingly, the at leastone Mie resonance in some embodiments may comprise at least one of thehigher order resonances.

In embodiments, the method is performed for nonlinear wavelengthconversion, and the input flux of pump energy comprises at least oneinput beam of electromagnetic radiation having a frequency f. The inputbeam is impinged onto the one or more dielectric optical resonators,thereby to generate converted light containing at least one convertedcomponent having a frequency unequal to f. The at least one Mieresonance of each of the dielectric optical resonators is excited by theinput beam. For example, a single input beam may be used to generate aharmonic or a harmonic series of the pump frequency f by a process ofharmonic generation. A “harmonic series” is a plurality of harmonics ofdifferent orders, such as second harmonic, third harmonic, etc.

In embodiments, the input beam comprises two beams which may bearbitrarily denominated a “pump” beam and a “signal” beam. The convertedlight is generated from the pump and signal beam by sum frequencygeneration, difference frequency generation, four-wave mixing, five-wavemixing, six-wave mixing, or some combination of these processes.

In embodiments, converted light is generated from a pump beam byspontaneous parametric down-conversion.

In embodiments, the nonlinear optical medium has a transition energy,and the converted light is generated from the pump beam by a process oftwo-photon emission with emitted photon energies that are less than thetransition energy. In embodiments, a quantum well multilayer embedded ina semiconductor host comprised by the nonlinear optical medium supportsat least one inter-subband transition, and the inter-subband transitionprovides the nonlinear optical medium transition energy.

In embodiments, a pump beam and a signal beam each excite a respectiveMie resonance of the one or more dielectric optical resonators.

In embodiments, the one or more dielectric optical resonators have atleast two Mie resonances, one said Mie resonance is excited by the inputbeam, and at least one other Mie resonance is excited by the at leastone converted component.

In embodiments, the at least one input beam comprises a pump beam offrequency f, the generating converted light comprises photoluminescencewhereby the converted light contains a spectrum of converted components,and the converted components in the photoluminescence spectrum have atleast some frequencies that are greater than the pump frequency f.

In embodiments, the input flux of pump energy is an electric current andthe nonlinear process is two-photon emission, or spontaneous parametricdown-conversion, or both two-photon emission and spontaneous parametricdown-conversion.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a theoretical transmission spectrum, obtained fromsimulations, of an array of cylindrical resonant structures formed inthe shape of circular cylinders. Our studies predict that arrays of suchresonators can exhibit enhanced optical properties, even withoutneighbor interactions or Fano resonances, when the resonator structuresare made with sub-wavelength dimensions. It will be seen that there areresonances at about 980 nm and about 1170 nm.

FIGS. 2 and 3 are theoretical two-dimensional contour plots, obtainedfrom computational simulations, of the electric field intensity within aresonator of the array of FIG. 1 at respective excitation wavelengths ofabout 980 nm and about 1170 nm.

FIG. 4 is a sketch, not to scale, of a shape for a resonator thatincludes an asymmetric feature effective for coupling of differentmodes. In the figure, the asymmetric feature is a cutout corner. Such ashape can be useful for producing resonator arrays that exhibit Fanoresonance.

FIG. 5 is a schematic diagram of a layered resonator structure in whichthe active medium comprises a quantum well multilayer, according to anembodiment of the invention.

FIG. 6 is a flowchart containing sequential, schematic cross-sectionalviews that illustrate a flip-chip method of assembling a resonant array.The illustrated method is useful when, e.g., in-situ oxidation toproduce an optical confinement layer is not feasible.

FIG. 7 is a notional diagram in which an overlying structure is added tothe layered resonator structure. The overlying structure includeselectrical contact layers and can be used for current injection orcurrent extraction. The term “overlying” as used here indicates arelative position only.

FIG. 8 is a notional perspective drawing of a unit cell of athree-dimensional resonant array in which three resonator structures areepitaxially grown in a stack.

FIG. 9 is a flowchart of a process for creating an array of galliumarsenide resonators, starting from a multilayer wafer.

FIG. 10 provides a view, based on a scanning electron microscope (SEM)image, of an example three-dimensional resonator array fabricatedaccording to methods described here.

FIG. 11 is a rendered 75° side-view SEM image of a fabricated GaAsdielectric resonator array. The GaAs resonators have the same diameterof about 250 nm and height of 300 nm.

FIG. 12 is a rendered top-view SEM image of the resonator array of FIG.11. The inset is the reflectivity spectrum of the GaAs resonator array.It will be seen that the spectrum exhibits two well-separatedreflectivity peaks corresponding respectively to the magnetic andelectric dipole resonances. The scale bars correspond to a length of 1μm.

FIG. 13 is a block diagram of an experimental setup for second-harmonicgeneration (SHG) measurements in a reflection geometry. The legends inthe figure have the following meanings: S, sample; O, objective; P,polarizer; BS, beam splitter. The inset shows that a coordinate systemis chosen in which the pump propagates along the z axis and the pumppolarization is along the x axis.

FIGS. 14 and 15 are graphs of experimental results showing the spectraldependence of the SHG intensity on linear and logarithmic scales,respectively. Resonantly enhanced SHG behavior is evident at themagnetic and electric dipole resonances. A background curve included inFIG. 14 is a simulated linear reflectivity spectrum of the sample. Abackground curve included in FIG. 15 is the corresponding experimentallymeasured spectrum.

FIG. 16 is a graph of experimental data showing a quadratic relationshipbetween the average pump and SHG powers at low pump intensities. Athigher pump intensities, there is seen a deviation from the quadraticrelationship that we attribute to damage to the GaAs resonators.

FIG. 17 is a graph of SHG conversion efficiency as a function of thepump power. The inset is an SEM image of damaged GaAs resonatorsresulting from illumination at a high average pump power of about 27 mW.

FIG. 18 is a block diagram of the experimental setup for afrequency-super-mixing study that we carried out using a femtosecondlaser system.

FIG. 19 is a reflectivity spectrum of a metasurface that wecharacterized in the study of FIG. 18. The two arrows seen in the figureindicate the two pump wavelengths.

FIG. 20 is a frequency-super-mixing spectrum that we obtainedexperimentally with a temporally overlapped pair of pump pulses in thestudy of FIG. 18. The spectrum displays a plurality of newly generatedfrequencies that have originated from various nonlinear opticalprocesses.

FIG. 21 is a scatterplot of measured versus predicted wavelengthsgenerated by a fifth-order nonlinear optical mixing effect with tuningof the pump wavelengths. The plot confirms that there was 4ω₂−ω₁frequency mixing.

FIG. 22 is a spectrum of second, third, and fourth harmonics generatedin the study of FIG. 18 using a single pump beam. The inset shows azoom-in spectrum of the fourth harmonic.

DESCRIPTION

Optical metamaterials are structures that interact with electromagneticradiation through geometrical features whose spatial scale is comparableto the wavelength of the interacting visible, infrared, or microwaveradiation. These materials are currently the subject of intense interestbecause of their many potential applications in the modification andcontrol of optical and microwave signals.

As research on optical metamaterials progresses, new effects withpotentially useful applications continue to be discovered. For example,we recently found a surprising enhancement in the predicted confinementof the resonant electric field within the resonator. This result arosefrom computer modeling of metamaterial structures. In an exampledescribed below, the structures include gallium arsenide (GaAs) andaluminum gallium arsenide (AlGaAs) layers grown on a GaAs substrate.

In an example, the initial workpiece for forming a GaAs—AlGaAs opticalmetamaterial is a layered structure consisting of a GaAs substrateoverlain by 400 nm of epitaxially grown AlGaAs, followed by 300 nm ofepitaxially grown GaAs. The aluminum in the AlGaAs layer is oxidized toAlGaO to reduce the refractive index of that layer to a valuesignificantly below the refractive index of GaAs. (AlGaO is analuminum-gallium oxide. Stoichiometric coefficients are not providedhere because the precise composition may be variable.) That is, therefractive index of the native alumina produced in this manner is about1.6, whereas GaAs has an index greater than 3.0 at typical near-infraredwavelengths. This has the desirable effect of confining the opticalintensity profile (corresponding to the resonant optical modes) withinthe resonators.

The workpiece is patterned and etched down to the top of the oxidizedAlGaAs layer to form an array of cylinders formed with sub-wavelengthdimensions, e.g. a diameter and height of several hundred nanometers forperformance at 1-2 μm optical wavelength.

FIG. 1 is the theoretical transmission spectrum, obtained fromsimulations, of an array of resonant structures of the kind describedabove. It will be seen that there are resonances at about 980 nm andabout 1170 nm.

The enhancement in the electric field intensity within the resonatorstructure can be seen in FIGS. 2 and 3. FIG. 2 is a theoreticaltwo-dimensional contour plot, obtained from simulations, of the electricfield intensity within the resonator at about 980 nm. FIG. 3 is asimilar two-dimensional contour plot of the electric field intensitywithin the resonator at about 1170 nm. The internal field is higher thanthe incident field because of the quality factor of the magnetic andelectric dipole resonances of the structure.

In another example, the initial workpiece is patterned so as to achieveFano resonances, which characteristically have extremely narrow spectralwidths. These resonances are believed to arise because of couplingbetween the different dipole modes of the dielectric blocks thatconstitute the individual resonators, as will be explained below.Modeling studies predict that Fano-resonant metamaterials can bedesigned to exhibit not only the internal field enhancements describedabove, but also transmission and reflection spectra that have extremelynarrow features.

FIG. 4 is an example of a resonator structure designed to exhibit a Fanoresonance when replicated as a unit cell in a two-dimensional array. Byway of example, the structure shown in FIG. 4 is a cube from which acorner has been cut out. The cutout extends through the entire thicknessof the cube, and it has a length and width that are each one-half theside length of the cube. It should be noted that this example isnon-limiting. Various other dimensions, as well as various other shapes,may also be effective for producing useful Fano resonances.

The width, length and height dimensions of the resonator structure willtypically be the designed Fano wavelength divided by the refractiveindex, so that the spatial dimensions are all less than the vacuumwavelength. However, larger dimensions can also be useful whenexcitation at higher-order resonant modes is desired.

Intersubband transitions (ISTs) are useful for enhancing nonlinearoptical effects due to their giant nonlinear optical susceptibilities.ISTs are also useful for providing sub-bandgap transitions that canparticipate in the nonlinear process known as two-photon emission (TPE).One useful way to provide ISTs is to add a quantum well multilayer to ahost semiconductor.

FIG. 5 provides an example of a resonator structure that incorporates aquantum well multilayer.

In the example of FIG. 5, a multilayer of indium gallium arsenide(InGaAs) quantum wells is formed in a gallium arsenide host on a galliumarsenide substrate. Quantum well multilayers useful in the presentcontext can also be implemented in a variety of other material systems.One example uses indium gallium arsenide phosphide (InGaAsP) multilayerson an indium phosphide (InP) substrate. In such a layer, InGaAsP wellsin which the alloy composition is adjusted to provide a lower bandgapare alternated with InGaAsP barrier layers in which the alloycomposition is adjusted to provide a higher bandgap.

A significant feature of confinement-type structures such as quantumwell and quantum dot multilayers is that they often support opticaltransitions of lower energies than the bandgap of the host material. Inthe quantum well multilayer of FIG. 5, for example, the resonanttransitions due to the InGaAs quantum wells will typically havewavelengths longer than the wavelength associated with bandgap-energytransitions in the gallium arsenide host.

Accordingly, the resonator body can for some purposes be dimensioned tosupport resonance at one or more wavelengths associated with transitionsin a gain medium such as a quantum well multilayer, without supportingresonance at the shorter wavelength associated with the host band-gapenergy, where undesirable optical loss might occur.

With further reference to FIG. 5, it will be seen there that a layeredstructure consists of a GaAs substrate 10 overlain in sequence by 300 nmlayer 15 of AlGaAs, base layer 20 of approximately 100 nm of GaAs, aquantum well multilayer 25 of five periods of InGaAs quantum wells(QWs), and a cap layer 30 of GaAs approximately 100 nm thick.Two-dimensional arrays of metamaterial resonators can be fabricated inthis structure using standard lithographic processes. For operation inthe near infrared, example resonator structures are cylinders ormodified cubes or the like having lateral dimensions of several hundrednanometers.

The 100-nm base layer 20 of GaAs forms the lower half of the resonator.The quantum-well multilayer 25 forms the middle portion of theresonator, and the upper 100-nm cap layer 30 of GaAs forms the upperhalf of the resonator.

Each quantum-well period consists of one layer 26 of GaAs and one layer27 of InGaAs. The III-V layers are deposited by, e.g., MOCVD or MBE. Thelayer thicknesses are typically in the range 1-20 nm, but may be variedaccording to known principles in order to achieve desired effects. Theresonator structures can be defined by a standard lithographic processflow such as electron-beam patterning and development, followed by metaldeposition, lift-off, and dry etching. For longer operating wavelengths,e.g. mid-infrared wavelengths, photolithography can be used. The etchwill typically extend all the way to the surface of the GaAs substrate10, but it can optionally stop at the top of the 300-nm AlGaAs layer 15or within that layer.

In embodiments, it is advantageous for the 300-nm-thick AlGaAs layer 15to have a high aluminum mole fraction, for example a mole fractiongreater than 80%. The aluminum can be oxidized to alumina (i.e., inAlGaO as explained above) to reduce the refractive index of layer 15 toa value significantly below the refractive index of GaAs. This has thedesirable effect of confining the optical intensity profile(corresponding to the resonant optical modes) within the resonators.

Oxidation of AlGaAs is described, for example, in Kent D. Choquette etal., “Advances in Selective Wet Oxidation of AlGaAs Alloys”, IEEE J.Sel. Topics in Quant. Electr. Vol. 3, No. 3 (June 1997) 916-926, theentirety of which is hereby incorporated herein by reference. Briefly,the oxidation is a one-step process in which the samples are introducedto an oxidation furnace where several gases flow through the samples athigh temperature. Complete oxidation of the AlGaAs is effective forachieving the desired mode confinement. Simulations indicate thatpartial oxidation can also be effective.

In alternative material systems in which the oxidation step is notfeasible, similar optical confinement can be achieved using flip-chipattachment. A known method of flip-chip attachment is described, forexample, in the article S. Person et al., “Demonstration of Zero OpticalBackscattering from Single Nanoparticles,” Nano Letters 13 (2013)1806-1809. In the work reported there, an epitaxial lift-off techniquewas used in conjunction with a wafer-bonding procedure to attach a highquality GaAs membrane, which was grown on a gallium arsenide substrate,to a fused silica substrate. For that work, directly growing galliumarsenide on fused silica was disfavored because it would create a highdensity of dislocations.

In other examples of flip-chip attachment, assembly is by well-knownprocesses, using a thin layer of adhesive or other material to adherethe confinement substrate to the resonator substrate.

Methods of flip-chip attachment would be desirable, for example, when itis desired to layer a semiconductor structure over a low-index substratesuch as glass or sapphire.

FIG. 6 illustrates another method of flip-chip attachment. As seen inthe figure, an epitaxially grown starting wafer includes a substrate 31,an etch-stop layer 32, and an active region 33. A low-refractive indexsubstrate 34 is provided for use as the optical confinement layer andalso as a handle for the active region. Low-index substrate 34 isflip-chip bonded to the top of the starting wafer, i.e., to the face ofactive region 33 distal substrate 31. For the purpose of bonding, a thinadhesion layer may be added. Substrate 31 and etch-stop layer 32 areremoved from the assembly by polishing and chemical etching. Activeregion 33 is then patterned and etched to produce the desired resonatorstructures 35. For clarity of presentation, several steps have beenomitted from this discussion. Various procedures for the omitted stepsare conventional and will be known to those skilled in the art.

Turning back to FIG. 5, it should be noted that the base layer 20,described above as consisting of GaAs, can alternatively be composed ofAlGaAs having a low aluminum concentration. The precise mole fraction ofaluminum is not critical, provided that when the 300-nm-thick AlGaAslayer 15 is oxidized, the base layer will oxidize much more slowly. Thisis achievable because the rate of oxidation varies exponentially withthe aluminum concentration. Reference is made, in this regard, to FIG. 8of the paper by Kent D. Choquette et al. cited above.

The oxidative procedure described above and the process of flip-chipdescribed above are non-limiting examples of processes that be usefulfor providing an optical confinement layer that has a lower refractiveindex than the constituent materials of the array of dielectricresonators.

It is important to note that similar structures can be created in otherIII-V material systems such as InAs, InP, InSb, GaSb, and variousIII-nitrides, as well as in II-VI material systems. It is also importantto note that the multilayer dielectric resonator structures (both III-Vand II-VI) can be fabricated directly on CMOS-compatible siliconsubstrates, because the resonators have ultra-small lateral dimensionsand can accommodate the strain associated with the lattice mismatch.This provides one potential solution to the difficulties ofincorporating light sources onto silicon wafers.

FIG. 7 provides a schematic diagram illustrating an overlying structurethat can be used for current injection into the multilayer resonantstructures in order to energize them. Two-photon emission, for example,can be produced by an electrically driven process as well as by opticalpumping. Spontaneous parametric down-conversion can also be produced byan electrically driven process as well as by optical pumping.

In an example, the GaAs cap layer 30 overlying the multiple-quantum-well(MQW) active region 25 is doped n-type and the GaAs layer 20 lying belowthe MQW active region is doped p-type. Because refractive index contrastis needed to preserve the mode confinement in the resonator structure,it is advantageous to use a layer of indium tin oxide (ITO), which has arefractive index of 1.31 at a wavelength of 1 μm, for both the p-typecontacts 40 and the n-type contacts 45. A dielectric layer 50 of, e.g.,silicon oxide is interposed between the two ITO contact layers.

Arrow 55 in FIG. 7 indicates a direction of current injection, utilizingthe ITO contact layers, for operation as, e.g., a photo-emissive device.

Although the preceding discussion has been directed to two-dimensionalarrays of resonators, it should be understood that three-dimensionalarrays are also feasible and within the scope of the present invention.Using epitaxial growth techniques, columns can be fabricated in whichtwo or more resonator structures are stacked one above the other.

FIG. 8 provides an example of such a column. As seen in the figure, agallium arsenide substrate 60 is overlain by a column consisting ofbottom resonator 65, middle resonator 70, and top resonator 75. Each ofthe three resonators includes an AlGaAs base layer 80 and a GaAs activelayer 85. Although not shown in the figure, each GaAs active layer maybe a host layer that contains a quantum-well multilayer as describedabove.

Fabrication of GaAs Two-Dimensional Arrays

We fabricated GaAs two-dimensional dielectric resonator arrays using acombination of high-aspect-ratio etching and selective wet oxidation ofAlGaAs under-layers to form a low refractive index oxide, i.e. native(Al_(x)Ga_(1-x))₂O₃ having a refractive index of about 1.6. As explainedbelow, we used the same fabrication processes to demonstratethree-dimensional GaAs dielectric resonator arrays.

Our selective wet oxidation technique was adapted from a techniquepreviously reported for forming current-blocking layers invertical-cavity surface-emitting lasers. The earlier technique wasreported, e.g., in the paper by K. D. Choquette et al., cited above.

FIG. 9 shows the process flow for creating GaAs resonators starting froma wafer, grown by molecular beam epitaxy (MBE), that consisted of asemi-insulating GaAs substrate onto which a 300-nm layer ofAl_(0.85)Ga_(0.15)As was deposited followed by a 300 nm layer of GaAs.

At step 100, we deposited a few hundred nanometers of silicon dioxide(SiO₂) to use as an etch mask. At step 105, we next spin-coated apositive tone polymethyl methacrylate (PMMA) resist. At step 110, wepatterned circular disks using standard electron-beam lithography.

At step 115, after the development of the PMMA, a 10-20 nm layer ofnickel was deposited. At step 120, this was followed by a lift-offprocess resulting in thin nickel disks. At step 125, the shape of thenickel disks was transferred onto the SiO₂ layer usinginductively-coupled-plasma (ICP) etching. At step 130, the nickel diskswere then removed using nitric acid, leaving only the SiO₂ disks as anetch mask for GaAs. Silicon dioxide was desirable for use as an etchmask because under the chlorine-based ICP etch that we used in thefollowing step, silicon dioxide etches more than five times more slowlythan GaAs or AlGaAs.

At step 135, we used an optimized, chlorine-based ICP etch recipe tocreate pillars of GaAs and AlGaAs having smooth vertical side walls. Atstep 140, the workpiece was placed in a tube furnace at about 420° C.for selective wet oxidization of the AlGaAs layers. A nitrogen carriergas was used to transport water vapor across the sample, converting thelayers of AlGaAs into their native oxide, i.e., low-index(Al_(x)Ga_(1-x))₂O₃.

In a modified version of the process of FIG. 9, we used an etch mask ofhydrogen silsesquioxane (HSQ) in place of PMMA and silicon dioxide. Thiswas a simpler process that produced similar results to the process ofFIG. 9. In the modified process, steps 100 and 115-130 were omitted.Instead, HSQ instead of PMMA was deposited by spin coating at step 100.At step 110, we patterned circular disks in the HSQ using standardelectron-beam lithography. The rest of the process was as describedabove. The HSQ that we used was Fox® 16 Flowable Oxide from the DowCorning corporation. The thickness of the HSQ layer affects the etchdepth that can be achieved. At the viscosity of undiluted HSQ, wedeposited a layer of HSQ about 300 nm thick, which allowed an etch depthof about 3 μm.

It should be noted that the GaAs resonator layers 160 can be replacedwith AlGaAs layers, provided the aluminum concentration of the resonatorlayers is at least about 20% lower than that of the oxidation layers155. The oxidation rate increases exponentially with aluminumconcentration, hence if the aluminum concentration in resonator layer160 is low enough, that layer will remain mostly unchanged while theunderlayer is completely oxidized.

An AlGaAs dielectric resonator layer 160 has beneficial applicationsbecause it can provide Mie resonances in the visible spectrum. That is,Al_(0.45)Ga_(0.55)As has a direct bandgap at 624 nm, but it transitionsinto an indirect bandgap material when the aluminum concentrationexceeds 45%.

Our fabrication technique, including the oxidation step, can be appliedto other aluminum-containing semiconductors, such as aluminum indiumarsenide and aluminum indium gallium phosphide.

More information about our fabrication method and designs may beobtained from the article, Sheng Liu et al., “III-V semiconductornano-resonators-a new strategy for passive, active, and nonlinearall-dielectric metamaterials,” published online at arXiv:1605.00298[physics.optics] (2016), the entirety of which is hereby incorporatedherein by reference for all purposes.

We believe that the flip-chip methods described above will also beuseful for fabricating dielectric resonator arrays of GaAs and othermaterials.

Fabrication of Multilayer Arrays

As mentioned above, fabricating three-dimensional dielectricmetamaterials is possible using the techniques described here. By “threedimensional” structures we mean stacked multilayer structures (which aresometimes referred to as “quasi-three-dimensional”), as well as otherkinds of structures that repeat in the direction perpendicular to thesubstrate.

To demonstrate the ability to produce multilayer dielectric resonatorarrays, we fabricated monolithic arrays of columns on a GaAs substrate,in which each column contained three instances of the(Al_(x)Ga_(1-x))₂O₃—GaAs pattern described above in Example 1. That is,the sequence of layers in each column, from the bottom up, wasGaAs/(Al_(x)Ga_(1-x))₂O₃/GaAs/(Al_(x)Ga_(1-x))₂O₃/GaAs/(Al_(x)Ga_(1-x))₂O₃/GaAs.Our fabrication process was as described above, except that the initialwafer had three layers of GaAs separated by three layers of AlGaAs.

FIG. 10 provides a view, based on a scanning electron microscope (SEM)image, of an example of the resulting structure. Visible in each columnare three approximately 300-nm-thick GaAs resonator layers 200, threeapproximately 300-nm-thick oxidized layers 205, a GaAs layer 210 at thebase of each column, and the GaAs substrate 215. A scale bar in thefigure indicates an interval of 1 μm.

The columns had a high aspect ratio, with a height somewhat greater than2 μm and an average diameter of about 350 nm or somewhat more. Thestructure was fabricated with a single-step ICP etch, since both AlGaAsand GaAs are etchable under the same conditions. We believe that evengreater numbers of layers can be included in structures of this kind.The number of layers in the initial wafer is ultimately limited by thelimitations of epitaxial growth. Other limits are practical limitsimposed by the thickness of etch mask, the etching conditions, and thedemands on the structural strength of the semiconductor materials forsupporting the resulting high aspect ratio structures.

With further reference to FIG. 10, it will be seen that the columns areslightly tapered with a diameter of about 350 nm at the top and about370 nm at the bottom. We believe that by adjusting the etchingconditions, we can control the taper angle of the multilayer structureso that the gallium arsenide resonator layers at each respective levelwithin the columns can have a distinct diameter. This can lead to adifferent resonant wavelength for the resonators at each respectivelevel.

The measured reflectivity spectrum of the three-layer array of FIG. 10agreed well with the FDTD simulation and clearly indicated the presenceof the electric and magnetic dipole resonances seen in the single-layerarray and described above.

Nonlinear Optical Processes

Nanostructured dielectric Mie resonators of the kind described above areuseful for various types of wavelength conversion resulting fromnonlinear optical processes. Our simulation results show highefficiencies of second order nonlinear conversions including secondharmonic generation, sum frequency generation and difference frequencygeneration. The efficiencies are several orders of magnitude higher thanthe same processes using conventional nonlinear crystals such as BBO orLBO with the same thicknesses (or optical paths).

The high efficiency is also contributed by the high nonlinearcoefficients of III-V semiconductors that are about 100 times higherthan those of BBO or LBO.

It is known from the theory of Mie scattering that the lowest tworesonances of a dielectric body (idealized as an isolated dielectricsphere) are an electric dipole resonance and a magnetic dipoleresonance. Many higher-order electric and magnetic Mie resonances arealso possible.

We also found out that the highest second harmonic generation efficiencyoccurs when the second-harmonic frequency coincide with the higher orderMie resonances and the fundamental frequency coincide with the first twodipole resonances (magnetic and electric dipole resonances). This isparticularly true here and for other nonlinear processes when both thepump frequency and the converted frequency coincide with respectiveresonances.

Simulations also show that different resonances (or differentcombinations of two resonances in the case of sum and differencefrequency generation) should be used for optimizing differentsecond-order nonlinear processes.

Using the same nanostructure, we also expect high efficiency ofspontaneous parametric down conversion (SPDC). SPDC is useful forgenerating entangled photon pairs. As such, it has broad applications inquantum information science. We believe that because our nanostructuresoffer the advantage of ultra-compactness, they can be advantageously beused to replace bulk nonlinear crystals or waveguide structures (whichrequire waveguide coupling) as efficient entangled photon sources.

Using the same nanostructures, we also believe that spontaneoustwo-photon emission is achievable. Spontaneous two-photon emissioninvolves the simultaneous emission of two photons due to an electronictransition between two levels with different energy. It has beenobserved in optically pumped bulk gallium arsenide and in electricallydriven GaInP/AlGaInP quantum wells. Like spontaneous parametricdown-conversion, spontaneous two-photon emission is a fundamentalphysics process useful for generating entangled photons, among otherthings.

We believe that electrically driven spontaneous two-photon emission canbe achieved by incorporating quantum wells in the resonators. As iswell-known in the art, alternating quantum well layers having differentband gap energies can be fabricated within, e.g., a gallium arsenidelayer by adding other Group III or Group V elements such as indium tolower the band gap or phosphorus to raise it. Further, we believe thatthe emission can be spectrally tuned by varying the dimensions andcompositions of the quantum wells and concomitantly varying thedimensions of the nanostructures so as to tune the resonances.

In some embodiments, multilayer structures such as those described abovecan be used to further increase the conversion efficiencies of processessuch as second harmonic generation, sum frequency generation, differencefrequency generation and spontaneous two-photon emission.

As those skilled in the art will understand, an “in-material” wavelengthof electromagnetic radiation of a given frequency within a body is thevacuum wavelength divided by the effective refractive index of the bodyat the given frequency. As those skilled in the art will alsounderstand, a “Mie resonator” is a dielectric body whose dimensionssupport one or more resonant modes of electromagnetic radiation.Typically, the size of a Mie resonator in all three spatial dimensionsis at least one in-material wavelength. For supporting low-order modes,the resonator may be about one in-material wavelength in extent in twoor in all three spatial dimensions. For supporting higher-order modes,various spatial dimensions may be commensurate with a multiple of thein-material wavelength.

Design and fabrication of GaAs dielectric Metasurfaces.

We design the GaAs resonators to support Mie magnetic and electricdipole resonances at wavelengths longer (i.e., at lower energy) thanthose corresponding to the GaAs bandgap. The reason for this is to avoidabsorption and to place the excitation within the spectral range of ourfemtosecond Ti:sapphire laser. It should be noted that for the choice ofdipole resonant frequencies reported here, absorption will still occurat those second-harmonic wavelengths that are shorter than thewavelength corresponding to the GaAs bandgap.

At the lowest dipole resonances, the resonators have side dimensionsthat are roughly the in-material wavelength λ/n, where λ is the freespace wavelength and n the refractive index. For operation at higherresonances, side dimensions that are approximately a multiple of thein-material wavelength will generally be desirable.

Our nonlinear metasurface comprises a square lattice of GaAs nanodiskresonators lying on a low refractive index (Al_(x)Ga_(1-x))₂O₃ nativeoxide spacer layer that is formed by selectively oxidizing high-Alcontent Al_(x)Ga_(1-x)As layers.

The pitch of the resonator array is designed to minimize the interactionbetween neighboring GaAs resonators. This is desirable in order tominimize line broadening, i.e., spectral broadening of the resonances ofthe individual resonators due to mutual interaction. Line broadeningreduces the quality factor of the resonator and can decrease the amountof field enhancement.

In a practical resonator array, the resonators may be regarded as“substantially non-interacting” if mutual interaction between theresonators results in less than 2% line broadening and less than 2%frequency shifting of the resonances, relative to individual isolatedresonators. Under an even stricter standard, the resonators may beconsidered “non-interacting” if the distance between them is at leastthe minimal separation that gives optimal performance (other factorsheld constant) for non-linear frequency conversion.

For applications using non-interacting resonators, it is not criticallynecessary to the operation of the device for the resonators to bearranged in a regular lattice. However, a regular lattice may beadvantageous for the purpose of beam shaping, as well as for achieving ahigh geometrical coverage factor. Further, a regular lattice may berequired in some embodiments in order to establish the conditionsnecessary for dark modes to emerge in the modal structure of the array.Dark modes may be advantageous for some applications because theirresonances can exhibit very high quality factors, which can lead, inturn, to high field enhancement.

Design factors that are important for optimizing nonlinear behaviorinclude the intrinsic nonlinearity of the constituent materials, thefield enhancement achieved within the resonator, the modal volume overwhich enhancement is achieved, and the amount of overlap that isachieved between jointly excited modes or between excited and emissivemodes. In regard to the last of these factors, it will often bedesirable to maximize the overlap integral of the intrinsic second-ordernonlinear susceptibility tensor χ⁽²⁾.

We believe that substantial field enhancement and effective operationcan be achieved with resonators having any of various shapes. For aresonator having a columnar conformation with a vertical axisperpendicular to the substrate, we mean by “shape” the cross section ina horizontal plane, i.e., the plane parallel to the substrate. Shapesbelieved to be suitable include the square, non-equilateral rectangle,circle, ellipse, and L-shape (i.e., a square or rectangle with a cornercut away.)

The height and lateral dimensions of the resonators provide designvariables that may be useful for tuning the resonances, and inparticular for shifting the spectral positions of the magnetic modesrelative to the electric modes. This can be advantageous, for example,when designing for the pump frequency to coincide with one of a pair ofrespective electric and magnetic dipole modes, and for a desirednonlinearly generated frequency to coincide with the other. Likewise, itcan be advantageous when designing for two frequencies (e.g., a pumpwave and a signal wave whose frequencies are to be added) to coincidewith the respective electric and magnetic dipole modes. Engineering ofthe resonant frequency in this manner can also facilitate tailoring ofthe emission pattern of the nonlinearly generated output beam.

Fabrication.

An example fabrication sequence begins with the molecular beam epitaxialgrowth of a 300-nm-thick layer of Al_(0.85)Ga_(0.15)As followed by a300-nm-thick layer of GaAs on top of a semi-insulating (100)-orientedGaAs substrate. We spin-coat a negative tone hydrogen silsesquioxane(HSQ Fox-16) resist on the sample and pattern circular disks usingstandard electron-beam lithography that converts the HSQ to SiO_(x). Theunexposed HSQ is developed using tetramethylammonium hydroxide leavingapproximately 500-nm-tall SiO_(x) nano-disks as etch masks for GaAs.

The shape of the SiO_(x) nano-disks is then transferred onto the GaAsand AlGaAs layers using an optimized chlorine-basedinductively-coupled-plasma (ICP) etch recipe. Finally, the sample isplaced in a tube furnace at about 420 degrees Celsius for a selectivewet oxidization process that converts the layers of Al_(0.85)Ga_(0.15)Asinto its oxide (Al_(x)Ga_(1-x))₂O₃. The resulting oxide has a lowrefractive index of about 1.6.

The large refractive index contrast between the GaAs resonators and theunderlying oxide ensures well-defined Mie modes with tightly confinedelectromagnetic fields inside the resonators, as is needed for efficientnonlinear optical generation.

FIGS. 11 and 12 respectively show renderings of a 75° degree side-viewSEM image and a top-view SEM image of a metasurface consisting of anarray of GaAs resonators with diameters of about 250 nm and heights of300 nm. The side-view image shows clear contrast between the top SiO_(x)etch mask, the GaAs resonators in the middle, and the AlGaO nano-disksat the bottom. The etch masks are not removed since, due to the lowrefractive index of SiO_(x), they barely perturb the distribution orintensity of the electromagnetic fields within the GaAs resonators. Thesample has an array pitch of 600 nm resulting in a spacing of about 350nm between resonators. Consequently, the interaction between theneighboring resonators is negligible.

Resonantly Enhanced SHG in GaAs Resonators

GaAs is known to possess large second-order nonlinearities with d₁₄ ofabout 200 pm/V (picometers per volt). This value is much higher than inconventional nonlinear crystals such as β-barium borate (d₂₂ of about2.2 pm/V) and LiNbO₃ (d₃₁ of about 6 pm/V and d₃₃ of about 30 pm/V).However, efficient SHG using GaAs has been challenging due to thedifficulty in meeting phase-matching conditions for long crystals in thezinc-blende crystal structure which exhibits isotropic refractiveindices.

In addition, (100)-GaAs possesses only one non-zero χ⁽²⁾ tensor element(d₁₄), which restricts the choice of nonlinear optical devicegeometries. In the work described below, however, we found that due tothe subwavelength layer thicknesses, resonantly enhanced SHG can beobtained from our GaAs dielectric metasurfaces without any provision forphase matching.

We performed the SHG measurements in reflection geometry because the SHGwavelengths are above the bandgap of GaAs, hence the SH signal in thetransmission direction would be completely absorbed by the GaAssubstrate.

FIG. 13 shows the experimental setup for measuring reflected SHGintensities and polarizations. We define the coordinate axes as shown inthe figure: the sample surface is the x-y plane, and pump propagatesalong the z axis.

Our optical pump was a mode-locked tunable Ti: sapphire laser oscillatorthat produced horizontally polarized pulses with 80-MHz repetition-rateand approximately 120 fs pulse width. The pump beam was reflected by adichroic beam splitter and then focused to a spot diameter of about 6 μmon the sample using a 20× near-infrared objective.

The generated SH was collected by the same objective, then transmittedthrough the beam splitter and measured using either a power meter or anear-IR spectrometer. The polarization of the SHG was measured using alinear polarizer. The detection efficiency of the entire system wascalibrated using a broadband calibration lamp. To simplify the physicalinterpretation, we rotated the sample about the z-axis so that the pumppolarization (along the x axis) was parallel to the [010]-direction ofthe GaAs wafer.

FIGS. 14 and 15 show the SHG intensity on linear and logarithmic scales,respectively, as the pump wavelength is tuned while keeping the pumppower constant. The simulated and experimental linear reflectivityspectra are used as the backgrounds for FIGS. 14 and 15, respectively.The SHG power exhibits peaks in the vicinity of the magnetic (about 1020nm) and electric (about 890 nm) dipole resonances due theelectromagnetic field enhancements that occur at these resonances. Inthis regard, the “field enhancement” is the ratio of the field intensitywithin the resonator (at a point of interest) to the pump intensityincident on the resonator.

Indeed, electromagnetic simulations described below show that at theseresonances the field intensities are about thirty times stronger thanthe incident pump intensity. The SH signal obtained when the pumpcoincides with the magnetic and electric dipole resonances are,respectively, more than three orders of magnitude and more than oneorder of magnitude higher than the signal obtained when pumping atoff-resonant wavelengths.

We believe that the large difference in SHG intensities obtained at themagnetic and electric resonances is partly due to higher absorption ofGaAs at the shorter SH wavelength associated with the electric dipoleresonance, and partly due to different origins of the SHG response atthe two dipole resonances. We note that the peak SH signals at the twodipole resonances are even larger (about four orders of magnitude at themagnetic dipole resonance) than the SH signal obtained from unpatternedGaAs.

The power dependence of the SHG signal is shown in FIG. 16. Thismeasurement was performed for a pump wavelength of 1020 nm whichcorresponds to maximum SHG efficiency. The quadratic power relationshipwas maintained over a wide pump power range until irreversible damage ofthe GaAs resonators occurred at an average power of about 5 mW (peakintensity of about 1.5 GW/cm²) as shown in the inset of FIG. 16.

At about 11 mW average pump power excitation, the SHG power was seen todecrease continuously over time (as indicated by the black triangles)due to physical damage to the sample. The inset of FIG. 17 shows thesevere damage caused to the GaAs resonators after illumination by a muchhigher average power of 27 mW (peak intensity of ˜8.1 GW/cm²). Thisdamage was likely associated with two-photon-absorption by GaAs,followed by thermal damage due to increased free carrier absorptionenhanced by the high electric field intensity inside the resonators.This implies that scaling to higher pump powers would require thefabrication of larger resonators, so that the dipole resonances (and,hence the pump photon energy) can be tuned to below one-half the GaAsbandgap.

It should be noted that surface defects created during the process ofICP dry etch could increase the loss of GaAs and therefore contribute tothe damage.

FIG. 16 also shows an enhancement of about four orders of magnitude toSHG from the GaAs metasurface relative to SHG on unpatterned GaAs.

FIG. 17 shows that the SHG conversion efficiency increases as the pumppower increases, and that it reaches a maximum conversion efficiency ofabout 2×10⁻⁵ when the pump power is about 11.4 mW (peak intensity ofabout 3.4 GW/cm²). Before reaching the damage threshold of about 5 mW,the nonlinear coefficient is about 1.5×10⁻⁸W/W², which is roughly twoorders of magnitude higher than a recently published record high SHGefficiency using mode-matching plasmonic nanoantennas.

The SHG enhancements arising from the electromagnetic field enhancementsof the GaAs metasurfaces can be treated using the effective second-ordernonlinear susceptibility tensor of the metasurface:

$\begin{matrix}{\chi_{ijk}^{{(2)}{eff}} = {\frac{\chi_{mnp}^{(2)}}{V}{\int_{V}{{f_{m{(i)}}^{2\;\omega}\left( {x,y,z} \right)}{f_{n{(j)}}^{\omega}\left( {x,y,z} \right)}{f_{p{(k)}}^{\omega}\left( {x,y,z} \right)}{dV}}}}} & \;\end{matrix}$where χ_(mnp) ⁽²⁾ is the material's intrinsic second-order nonlinearsusceptibility, V is volume, f^(2ω) is the field enhancement at the SHwavelength, and f^(ω) is the field enhancement at the fundamentalwavelength.

In this case, the SHG power is proportional to χ_(ijk) ^((2)eff)·I_(p)², where I_(p) is the incident pump intensity. Therefore, it isimportant to achieve high electromagnetic field enhancements at both thefundamental and SH wavelengths. However, our simulations show weakelectromagnetic fields inside the resonators at the SH wavelengths dueto the large absorption of GaAs at visible wavelengths. This tends tolimit the SHG efficiency that we can achieve.

Simulations also show that the electromagnetic fields are much weaker atthe SH of the electric dipole wavelength than at the SH of the magneticdipole wavelength, which partly explains the large difference betweenthe SHG powers obtained when pumping at the two dipole resonances.

Therefore, we expect substantially greater SHG conversion efficiency inlarger GaAs resonators designed for the magnetic dipole resonance tofall at wavelengths longer than twice the GaAs bandgap wavelength, or inresonators in which GaAs is replaced by the higher-bandgap AlGaAs.

It should be noted in this regard that in general, the aluminum fractionx in Al_(x)Ga_(1-x)As is limited as a practical matter to about 0.5because of the tendency of AlGaAs with greater aluminum content tooxidize in air.

Those changes would reduce absorption at the SH wavelength, therebypermitting greater electromagnetic field enhancements at the SHwavelength, and they would reduce the damage induced by two-photonabsorption at larger pump powers. Further improvements could also beobtained by optimizing the resonator shape in order to obtain a maximummodal overlap between the SH and fundamental wavelengths.

Absorption of the converted light by the substrate is a potentialdetriment when a device is operated in transmission instead ofreflection. As those skilled in the art will understand, the convertedlight is most susceptible to absorption if the photon energy exceeds thesubstrate bandgap, and is least susceptible to absorption if the photonenergy falls below the substrate bandgap. Hence even if the substrate istransparent to downconverted light, the second (or higher) orderharmonic light may be substantially attenuated. For such applications,it will be desirable to employ, for example, a flip-chip-basedfabrication technique in which the substrate is removed.

Our demonstration paves the way for using dielectric metasurfaces inother phase-matching free nonlinear optical applications such asnext-generation nonlinear optical converters for frequency mixing,photon pair generation, and all-optical-optical control and tunability.

For example, we believe that the resonant nanostructures described herecan be used for highly efficient spontaneous parametric down conversionto generate entangled photon pairs. In a sense, the down-conversionprocess is the inverse of the second-harmonic generation process. Thatis, examples of down-conversion will produce optical output containing asub-harmonic of the pump, with a photon energy that is one-half thephoton energy of the pump.

Example: Frequency-Super-Mixing in Dielectric Metasurfaces

We fabricated a square array of GaAs nanodisk resonators, beginning withmolecular beam epitaxial growth of a 400-nm-thick layer ofAl_(0.85)Ga_(0.15)As followed by a 450-nm-thick layer of GaAs.Electron-beam lithography and dry etching were used to define themetasurface geometries. Optical isolation was provided by AlGaO nativeoxide as described above.

To study frequency mixing in the GaAs metasurface sample we used twonear-infrared femtosecond beams that could independently be spectrallytuned (FIG. 18). FIG. 18 shows the experimental setup. As seen in thefigure, two pump beams B1, B2 are generated by an amplified Ti:sapphirelaser (1-kHz repetition rate) pumping atwin-optical-parametric-amplifier. As also seen in the figure, a delayline adds a variable relative delay between the pump beams before theyare combined by dichroic beam combiner Di. Near-IR objective O focusesthe combined beams onto the sample and collects reflected light. Thereflected light is directed via 1064-nm long pass filter LP into aspectrometer and directed via glass window G into a CCD camera.

The average powers of the pump beams incident on the metasurfaces wereabout 10 μW. The focal spots of the pumps were about 40 μm in diameter.For this measurement, we used a metasurface consisting of resonatorswith diameters of about 420 nm that supported magnetic and electricdipole resonances at about 1520 nm and about 1250 nm, respectively.

FIG. 19 is the measured reflectivity spectrum of the metasurface. Arrowsin the figure indicate the spectral locations of the two pump beams (λ₁of about 1220 nm and λ₂ of about 1570 nm). The two reflectivity peaksvisible in the figure correspond to the two dipole resonances.

It will be thus be understood from the figure that each of the pumpwavelengths was made to overlap with a respective one of the two dipoleresonances. That is, each of the pump center wavelengths fell within thebandwidth, at half-maximum, of one of the dipole resonances. We foundthat this was a factor in optimizing the frequency mixing signal.

To further enhance the conversion efficiency, it may be desirable tooptimize the modal overlap between the two resonances. This can be doneby varying the resonator geometry, or by introducing mutual interactionbetween neighboring resonators by reducing the resonator-to-resonatorseparation.

FIG. 20 shows the measured spectrum of frequency-super-mixing in theGaAs metasurface when the two pump pulses spatially and temporallyoverlap. We observed a total eleven spectral peaks spanning the spectrumfrom the UV to the near-infrared; the generation mechanism for each ofthe peaks is notated in the figure. The meanings of the legends in thefigure are: SHG, second harmonic generation; THG, third harmonicgeneration; FHG, fourth harmonic generation; SFG, sum frequencygeneration; FWM, four-wave mixing; PL, photoluminescence. We identifiedand confirmed these mechanisms by matching the photon energies and bymeasuring the power dependence.

Among these peaks, the one centered near 870 nm corresponds to thephotoluminescence of GaAs induced by two-photon-absorption of the pumpbeams. This was confirmed by a control experiment observingphotoluminescence from an unpatterned GaAs substrate.

We categorize the rest of the newly generated frequencies into twogroups: (i) those relying on only one of the two pump beams, such asthose due to the harmonic processes SHG, THG and FHG (inset of FIG. 22);and (ii) those resulting from the coincidence of both pump beams at thesame delay time, such as those due to SFG and FWM as well as otherthird- and fifth-order nonlinear frequency mixing processes.

We performed a study of the delay-dependent nonlinear frequency mixingspectra. We found that as expected, the harmonic generation processesrelying on a single pump beam were observed regardless of the delay. Incontrast, the signals relying on frequency-mixing processes such as SFG,FWM (2ω₁−ω₂), 2ω₁+ω₂, and ω₁+2ω₂ appeared only when the two pump pulses(also referred to herein as “pump” pulse and “signal” pulse) reached themetasurface simultaneously.

Additionally, we observed a fifth-order nonlinear optical mixing effectcorresponding to 4ω₂−ω₁. We confirmed this process by tuning thewavelengths of the two pumps. Our results are shown in FIG. 21, which isa scatterplot of the measured versus the expected output wavelength forvarious pump tunings. The pump-wavelength pairs (λ₁, λ₂) in nanometersfor the five data points plotted in the figure are, from bottom to top:(1248.4, 1557.6), (1234.8, 1558.6), (1211.4, 1558.8), (1234.8, 1581.2),and (1233.6, 1601).

As expected, the harmonic generation processes were the only nonlinearprocesses that we observed when the excitation came from a single pumpbeam only. This is apparent from the spectrum shown in FIG. 22, in whichonly the first, second, and third harmonics are evident. (The inset inthe figure shows a zoom-in of the fourth harmonic.) Higher harmonicgeneration can be expected if AlGaAs is substituted for GaAs to reducethe material absorption, and if the experimental setup is optimized forthe UV spectral range.

Based on the results reported above, we believe that our techniques arepotentially useful for performing optical frequency conversion by avariety of nonlinear optical processes. Using a single pump beam, theseprocesses include second harmonic conversion and higher-order harmonicconversion up to the fifth harmonic or even higher. Using two or morepump beams (for example, using a pump beam and a signal beam), theseprocesses include sum frequency conversion and the related process ofdifference frequency conversion, as well as four-wave mixing. We alsobelieve that higher-order frequency mixing processes can be achieved,such as five-wave mixing and six-wave mixing.

Also included are spontaneous two-photon emission, both by opticalpumping and by electrical pumping. We believe that using our resonatorarrays, the inverse process, i.e. two-photon absorption, is alsoachievable and that it may be useful for generating photoluminescence atconverted frequencies.

We also believe that using the same techniques, high efficiencies ofspontaneous parametric down-conversion (SPDC) will be achievable. SPDCis especially useful as a path for generating entangled photon pairs.

Another useful application of our technique would be for down-convertingor up-converting frequency combs using Mie resonators. This could beused to extend the frequency combs to other spectral regimes.

Another nonlinear phenomenon is supercontinuum generation, in whichseveral nonlinear process cooperate to severely broaden the spectrum ofa pump beam to produce a smooth spectral continuum. In addition to thevarious applications described above, we believe that our resonators arepotentially useful for supercontinuum generation.

We claim:
 1. A method of nonlinear wavelength generation using anonlinear optical medium and an input flux of pump energy comprising aninput beam of electromagnetic radiation having a frequency f, the methodcomprising: impinging the input beam onto one or more dielectric opticalresonators, each having an optical cavity comprising the nonlinearoptical medium, thereby to generate converted light containing at leastone converted component having a frequency attainable only through anon-linear process; and collecting, from the one or more dielectricresonators, a beam of output light comprising the at least one convertedcomponent; wherein: each of the one or more dielectric opticalresonators has at least one Mie resonance that is excited by the inputbeam; the at least one Mie resonance of each of the dielectric opticalresonators is excited by the input beam; the generating converted lightcomprises a process of harmonic generation, and the at least oneconverted component comprises a second harmonic of the input beamfrequency f.
 2. The method of claim 1, wherein the input flux of pumpenergy is applied to an array of nominally identical dielectric opticalresonators.
 3. The method of claim 1, wherein the at least one Mieresonance comprises a magnetic dipole resonance or an electric dipoleresonance or both an electric dipole resonance and a magnetic dipoleresonance.
 4. The method of claim 1, wherein each of the one or moredielectric optical resonators has an electric dipole resonance, amagnetic dipole resonance, and resonances of higher order than theelectric and magnetic dipole resonances, and wherein the at least oneMie resonance comprises at least one of the higher order resonances. 5.The method of claim 1, wherein the at least one Mie resonance that isexcited by the input beam comprises an electric dipole resonance, or amagnetic dipole resonance, or both an electric dipole resonance and amagnetic dipole resonance.
 6. A method of nonlinear wavelengthconversion using a nonlinear optical medium, one or more dielectricoptical resonators, each having an optical cavity comprising thenonlinear optical medium, and at least one input beam of electromagneticradiation having a frequency f, comprising: impinging the at least oneinput beam onto the one or more dielectric optical resonators so as toexcite at least one Mie resonance of each of the one or more dielectricoptical resonators, thereby to generate converted light containing atleast one converted component having a frequency unequal to f; andcollecting, from the one or more dielectric resonators, a beam of outputlight comprising the at least one converted component; wherein: the atleast one input beam comprises a pump beam and a signal beam, and thegenerating converted light comprises at least one process from the groupconsisting of sum frequency generation, difference frequency generation,four-wave mixing, five-wave mixing, and six-wave mixing.
 7. A method ofnonlinear wavelength conversion using a nonlinear optical medium, one ormore dielectric optical resonators, each having an optical cavitycomprising the nonlinear optical medium, and at least one input beam ofelectromagnetic radiation having a frequency f, comprising: impingingthe at least one input beam onto the one or more dielectric opticalresonators so as to excite at least one Mie resonance of each of the oneor more dielectric optical resonators, thereby to generate convertedlight containing at least one converted component having a frequencyunequal to f; and collecting, from the one or more dielectricresonators, a beam of output light comprising the at least one convertedcomponent; wherein: the at least one input beam comprises a pump beam,and the generating converted light comprises a process of spontaneousparametric down-conversion.
 8. A method of nonlinear wavelengthconversion using a nonlinear optical medium, one or more dielectricoptical resonators, each having an optical cavity comprising thenonlinear optical medium, and at least one input beam of electromagneticradiation having a frequency f, comprising: impinging the at least oneinput beam onto the one or more dielectric optical resonators so as toexcite at least one Mie resonance of each of the one or more dielectricoptical resonators, thereby to generate converted light containing atleast one converted component having a frequency unequal to f; andcollecting, from the one or more dielectric resonators, a beam of outputlight comprising the at least one converted component; wherein: thenonlinear optical medium comprises a semiconductor host and a quantumwell multilayer embedded in the semiconductor host.
 9. A method ofnonlinear wavelength conversion using a nonlinear optical medium, one ormore dielectric optical resonators, each having an optical cavitycomprising the nonlinear optical medium, and at least one input beam ofelectromagnetic radiation having a frequency f, comprising: impingingthe at least one input beam onto the one or more dielectric opticalresonators so as to excite at least one Mie resonance of each of the oneor more dielectric optical resonators, thereby to generate convertedlight containing at least one converted component having a frequencyunequal to f; and collecting, from the one or more dielectricresonators, a beam of output light comprising the at least one convertedcomponent; wherein: the at least one input beam comprises a pump beam,the nonlinear optical medium has a transition energy, and the generatingconverted light comprises a process of two-photon emission with emittedphoton energies that are less than the transition energy.
 10. The methodof claim 9, wherein the nonlinear optical medium comprises asemiconductor host and a quantum well multilayer embedded in thesemiconductor host, the quantum well multilayer supports at least oneinter-subband transition, and the inter-subband transition provides thenonlinear optical medium transition energy.
 11. A method of nonlinearwavelength conversion using a nonlinear optical medium, one or moredielectric optical resonators, each having an optical cavity comprisingthe nonlinear optical medium, and at least one input beam ofelectromagnetic radiation having a frequency f, comprising: impingingthe at least one input beam onto the one or more dielectric opticalresonators so as to excite at least one Mie resonance of each of the oneor more dielectric optical resonators, thereby to generate convertedlight containing at least one converted component having a frequencyunequal to f; and collecting, from the one or more dielectricresonators, a beam of output light comprising the at least one convertedcomponent; wherein: the at least one input beam comprises a pump beamand a signal beam, and the pump beam and the signal beam each excite arespective Mie resonance of the one or more dielectric opticalresonators.
 12. A method of nonlinear wavelength conversion using anonlinear optical medium, one or more dielectric optical resonators,each having an optical cavity comprising the nonlinear optical medium,and at least one input beam of electromagnetic radiation having afrequency f, comprising: impinging the at least one input beam onto theone or more dielectric optical resonators so as to excite at least oneMie resonance of each of the one or more dielectric optical resonators,thereby to generate converted light containing at least one convertedcomponent having a frequency unequal to f; and collecting, from the oneor more dielectric resonators, a beam of output light comprising the atleast one converted component; wherein: the one or more dielectricoptical resonators have at least two Mie resonances, one said Mieresonance is excited by the input beam, and at least one other Mieresonance is excited by the at least one converted component.
 13. Amethod of nonlinear wavelength generation using a nonlinear opticalmedium and an input flux of pump energy, comprising: applying the inputflux of pump energy to one or more dielectric optical resonators, eachhaving an optical cavity comprising the nonlinear optical medium,thereby to generate converted light containing at least one convertedcomponent having a frequency attainable only through a non-linearprocess; and collecting, from the one or more dielectric resonators, abeam of output light comprising the at least one converted component;wherein: each of the one or more dielectric optical resonators has atleast one Mie resonance that is excited by the input flux of pumpenergy; and the input flux of pump energy is an electric current and thenonlinear process is two-photon emission, or spontaneous parametricdown-conversion, or both two-photon emission and spontaneous parametricdown-conversion.